Prove that √5 is irrational

Prove that √5 is irrational

Dear Student

Let us assume, that√5is rational number.
i.e.√5= x/y ( where x and y are co primes )
y√5= x
Squaring both the sides, we get,
(y√5)^2= x^2
⇒5y2= x2...................................... (1)
Thus, x2isdivisible by 5, so x is also divisible by 5.
Let us say, x = 5k, for some value of k and substituting the value of x in equation (1), we get, 5y^2= (5k)^2
⇒y^2= 5k^2
𝒚^𝟐is divisible by 5 it means y is divisible by 5.
Therefore, x and y are co-primes.Since, our assumption about√5is rational is incorrect.
Hence,√5is irrational number.

​Thanks